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Hilbert

Syntax

Hilbert(R:RING or TAGGED("Quotient")):TAGGED("$hp.Hilbert")
Hilbert(R:RING or TAGGED("Quotient"),N:INT):INT
    

Summary

the Hilbert function

Description

The first form of this function computes the Hilbert function for R. The second form computes the N-th value of the Hilbert function. The weights of the indeterminates of R must all be 1. If the input is not homogeneous, the Hilbert function of the corresponding leading term (initial) ideal or module is calculated. For repeated evaluations of the Hilbert function, use EvalHilbertFn instead of Hilbert(R,N) in order to speed up execution.

The coefficient ring must be a field.

example

    
Use R ::= Q[t,x,y,z];
Hilbert(R/Ideal(z^2-xy,xz^2+t^3));
H(0) = 1
H(1) = 4
H(t) = 6t-3   for t >= 2
-------------------------------
M := R^2/Module([x^2-t,xy-z^3],[zy,tz-x^3y+3]);
Hilbert(M);
H(0) = 2
H(1) = 8
H(2) = 20
H(3) = 39
H(t) = 3t^2 + 6t-7   for t >= 4
-------------------------------
Hilbert(M,3)
39
-------------------------------
Hilbert(M,5);
98
-------------------------------
        
    

See Also